This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A188570 #44 Oct 25 2024 09:34:19
%S A188570 1,1,6,16,80,296,1296,5216,21952,90304,375936,1555456,6456320,
%T A188570 26754560,110963712,460015616,1907494912,7908659200,32792076288,
%U A188570 135963148288,563742310400,2337417887744,9691567030272,40183767891968,166612591968256,690819710058496
%N A188570 a(n) = coefficient of the term that is independent of sqrt(2) and sqrt(3) in the expansion of (1 + sqrt(2) + sqrt(3))^n.
%C A188570 From _Clark Kimberling_, Oct 23 2024: (Start)
%C A188570 Conjecture: every prime divides a(n) for infinitely many n, and if K(p) = (k(1), k(2),...) is the maximal subsequence of indices n such that p divides a(n), then the difference sequence of K(p) is eventually periodic; indeed, K(p) is purely periodic for the first 6 primes, with respective period lengths 1,5,5,3,3,5 and these periods:
%C A188570 p = 2: (4)
%C A188570 p = 3: (8, 1, 4, 3, 8)
%C A188570 p = 5: (9, 10, 1, 20, 20)
%C A188570 p = 7: (9, 9, 18)
%C A188570 p = 11: (10, 11, 43)
%C A188570 p = 13: (7, 21, 21, 7, 28)
%C A188570 See A377109 for a guide to related sequences. (End)
%H A188570 Vincenzo Librandi, <a href="/A188570/b188570.txt">Table of n, a(n) for n = 0..200</a>
%H A188570 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,-16,8).
%F A188570 Recurrence: a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3) + 8*a(n-4). - _Vaclav Kotesovec_, Aug 13 2013
%F A188570 a(n) ~ (1+sqrt(3)+sqrt(2))^n/4. - _Vaclav Kotesovec_, Aug 13 2013
%e A188570 a(3) = 16 because (1+sqrt(2)+sqrt(3))^3 = 16 + 14*sqrt(2) + 12*sqrt(3) + 6*sqrt(6).
%t A188570 a[n_] := Sum[Sum[2^(Floor[n/2] - k - j) 3^j Multinomial[2 k + n - 2 Floor[n/2], 2 j, 2 Floor[n/2] - 2 k - 2 j], {j, 0, Floor[n/2] - k}], {k, 0, Floor[n/2]}]; Table[a[n], {n, 0, 25}]
%t A188570 a[n_] := Expand[(1 + Sqrt[2] + Sqrt[3])^n] /. Sqrt[_] -> 0; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jan 08 2013 *)
%t A188570 LinearRecurrence[{4,4,-16,8},{1,1,6,16},30] (* _Harvey P. Dale_, Jan 25 2019 *)
%Y A188570 Cf. A188571, A188572, A188573, A377109.
%K A188570 nonn
%O A188570 0,3
%A A188570 Mateusz Szymański, Dec 28 2012
%E A188570 Edited by _Clark Kimberling_, Oct 20 2024